Lower bounds for fluctuations in first-passage percolation for general distributions

Abstract

In first-passage percolation (FPP), one assigns i.i.d. weights to the edges of the cubic lattice Z d and analyzes the induced weighted graph metric. If T (x, y) is the distance between vertices x and y, then a primary question in the model is: what is the order of the fluctuations of T (0, x)? It is expected that the variance of T (0, x) grows like the norm of x to a power strictly less than 1, but the best lower bounds available are (only in two dimensions) of order log x . This result was found in the '90s and there has not been any improvement since. In this paper, we address the problem of getting stronger fluctuation bounds: to show that T (0, x) is with high probability not contained in an interval of size o(log x ) 1/2 , and similar statements for FPP in thin cylinders. Such statements have been proved for special edge-weight distributions, and here we obtain such bounds for general edge-weight distributions. The methods involve inducing a fluctuation in the number of edges in a box whose weights are of "hi-mode" (large).

Fluctuation bounds for first-passage percolation on the square, tube, and torus

“…Our main goal is to find lower bounds for the fluctuations of these variables. We use the following definition of fluctuations, similar to that taken in [8].…”

“…A logarithmic lower bound for the variance of the point-to-point minimal passage time T (x, y) in the standard model (FPP on the infinite discrete lattice Z 2 ) is well-known for a large class of distributions of edge weights; see [4,Sec. 3.3] and [5,8,11] for fluctuation bounds. A lower bound of polynomial order can even be shown under the curvature assumption we make below.…”

“…This reduces the problem to finding the order of fluctuations of n 1−α many independent cylinder passage times. Cylinder times have been studied in [1,6,8], but those works only provide lower bounds for the fluctuations of order n (1−α)/2 for a single time. To extend this to a minimum of many cylinder times, we need much more precise information about their distributions, in particular estimates for their 1/n 1−α quantile.…”

Geodesics Toward Corners in First Passage Percolation

“…Various authors have dealt with this situation in different ways. One approach is to prove far-fromoptimal (yet still difficult) results, for example that the variance of passage times over distance n is at least of order log n [13], or that for fixed k, for a special class of passage time distributions, the limit shape cannot be a polygon with fewer than k sides [17], both for d = 2. A second approach, taken in [2], [3], [29], and for non-integrable LPP in [21], is to prove conditional results, assuming certain fundamental unproven properties like (3) and/or (4) above, and showing that more delicate properties follow from them.…”